Devlin's
Angle

November 2004

Election Math

A quick glance at the political map of the United States
that appeared in most newspapers the morning after the US
General Election decries the oft repeated claim that the
people have decided. Both the map and the figures behind
them show that the people, taken as a whole, are about as
undecided as it is possible to be. Far from being united,
our nation is now cleanly divided into two regions having
very different political flavors, with the inland and
southern states all Republican (colored red) and the west
coast, north east, and north central regions all Democrat
(colored blue).

I am hardly the first person to observe that (regardless of
which candidate or party wins) it is quite ridiculous for a
nation as large and powerful as the USA, whose founding
ideals are built around the notion that the elected leaders
represent the entire population, to have its most
fundamental political choices come down to a few thousand
voters in one state (Ohio this time round, Florida in
2000).

The culprit is not the Electoral College, as some have
suggested, but the way we do the math of elections -- how
we count the vote. If we really believed in the intentions
of our Founding Fathers, we would use our intellectual
talent to devise an electoral tally system that truly
reflects the wishes of the electorate.

The electoral math used in the United States election
process counts votes using a system known as plurality
voting. In this system, also known as "first-past-the-
post," the candidate with the most votes is declared the
winner. This system has several major flaws. The most
obvious one is the one I alluded to above, where the latest
election has left 48% of the nation with a president they
do not like, do not trust, and whose personal beliefs make
it impossible for him to represent the views of many of the
very citizens he is supposed to lead. To anyone who truly
believes that the United States should in fact be
united, that we should be "One Nation," that kind of
outcome alone should be reason to seek a better system.

Another flaw with plurality voting, although this one did
not cause any problems in the latest presidential election,
is that the method can result in the election of a
candidate whom almost two-thirds of voters detest.

For instance, in 1998, in a three-party race, plurality
voting resulted in the election of former wrestler Jesse
Ventura as Governor of Minnesota, despite the fact that
only 37% of the electors voted for him. The almost two-
thirds of electors who voted Democrat or Republican had to
come to terms with a governor that none of them wanted --
or expected. Judging by the comments immediately after the
election, the majority of Democrat and Republican voters
were strongly opposed to Reform Party candidate Ventura
moving into the Governor's mansion. In which case, he won
not because the majority of voters chose him, but because
plurality voting effectively thwarted the will of the
people. Had the voters been able to vote in such a way
that, if their preferred candidate were not going to win,
their preference between the remaining two could be
counted, the outcome could have been quite different.

Several countries, among them Australia, the Irish
Republic, and Northern Ireland, use a system called
single transferable vote. Introduced by Thomas Hare
in England in the 1850s, this system takes account of the
entire range of preferences each voter has for the
candidates. All electors rank all the candidates in order
of preference. When the votes are tallied, the candidates
are first ranked based on the number of first-place votes
each received. The candidate who comes out last is dropped
from the list. This, of course, effectively
"disenfranchises" all those voters who picked that
candidate. So, their vote is automatically transferred to
their second choice of
candidate -- which means that their vote still counts. Then
the process is repeated: the candidates are ranked a second
time, according to the new distribution of votes. Again,
the candidate who comes out last is dropped from
the list. With just three candidates, this leaves one
candidate, who is declared the winner. In a contest with
more than three candidates, the process is repeated one or
more additional times until only one candidate remains,
with that individual winning the election. Since each voter
ranks all the candidates in order, this method ensures that
at every stage, every voter's preferences among the
remaining candidates is taken into account.

An alternative system that avoids the kind of outcomes of
the 1998 Minnesota Governor's race is the Borda
count, named after Jean-Charles de Borda, who devised
it in 1781. Again, the idea is to try to take account of
each voter's overall preferences among all the candidates.
As with the single transferable vote, in this system, when
the poll takes place, each voter ranks all the candidates.
If there are n candidates, then when the votes are
tallied, the candidate receives n points for each
first-place ranking, n-1 points for each second
place ranking, n-2 points for each third place
ranking, down to just 1 point for each last place ranking.
The candidate with the greatest total number of points is
then declared the winner.

Yet another system that avoids the Jesse Ventura phenomenon
is approval voting. Here the philosophy is to try to
ensure that the process does not
lead to the election of someone whom the majority opposes.
Each voter is allowed to vote for all those candidates of
whom he or she approves, and the candidate who gets the
most votes wins the election. This is the method used to
elect the officers of both the American Mathematical
Society and the Mathematical
Association of America.

To see how these different systems can lead to very
different results, let's consider a hypothetical scenario
in which 15 million voters go to the polls in an election
with three candidates, A, B, C. Their preferences between
the three candidates are as follows:

6 million rank A first, then B, then C.

5 million rank C first, then B, then A.

4 million rank B first, then C, then A.

If the votes are tallied by the plurality vote -- the
present system -- then A's 6 million (first-place) votes
make him the clear winner. And yet, 9 million voters (60%
of the total) rank him dead last! That hardly seems
fair.

What happens if the votes are counted by the single
transferable vote system -- the system used in Australia
and Ireland? The first round of the tally process
eliminates B, who is only ranked first by 4 million voters.
Those 4 million voters all have C as their second choice,
so in the second round of the tally process their votes are
transferred to C. The result is that, in the second round,
A gets 6 million first place votes while C gets 9 million.
Thus, C wins by a whopping 9 million to 6 million
margin.

But wait a minute. Looking at the original rankings, we see
that 10 million voters prefer B to C -- that's 66% of the
total vote. Can it really be fair for such a large majority
of the electorate to have their preferences ignored so
dramatically?

Thus, both the plurality vote and single transferable vote
can lead to results that run counter to the overwhelming
desires of the electorate. What happens if we use the Borda
count? Well, with this method, A gets

6m x 3 + 5m x 1 + 4m x 1 = 27m points,

C gets

6m x 1 + 5m x 3 + 4m x 2 = 29m points,

and B gets

6m x 2 + 5m x 2 + 4m x 3 = 34m points.

The result is a decisive win for B, with C coming in second
and A trailing in third place.

What happens with approval voting? Well, as I have set up
the problem so far, we don't have enough information -- we
don't know how many electors actively oppose each
particular candidate. Let's assume that C's supporters
and B's supporters could live with the others' candidate,
but the voters in both groups really don't want to see A
elected. In this case, B gets 15 million
votes, C gets 9 million votes, and A gets a mere 6 million.
All in all, it's beginning to look as though B is the one
who should win.

Another approach is to choose the individual who would beat
every other candidate in head-to-head, two-party contests.
This method was suggested
by the Marquis de Condorcet in 1785, and as a result is
known today as the Condorcet system.

For the scenario in our example, B also wins according to
the Condorcet system. He gets at least 10 million votes in
a straight B-C contest and at least 9
million votes in a A-B match-up, in either case a majority
of the 15 million voters. Unfortunately, although it works
for this example, and despite the fact that it has
considerable appeal, the Condorcet method suffers from a
major disadvantage: it does not always produce a clear
winner!

For example, suppose the voting profile were as follows:

5 million rank A first, then C, then B.

5 million rank C first, then B, then A.

5 million rank B first, then A, then C.

Then 10 million voters prefer A to C, so A would easily win
an A-C battle. Also, 10 million voters prefer C to B, so C
would romp home in a B-C contest. The remaining two-party
match-up would pit A against B. But when we look at the
preferences, we see that 10 million people prefer B to A,
so B comes out on top in that contest. In other words,
there is no clear winner. Each candidate wins one of the
three possible two-party battles!

One worrying problem with the single transferable vote is
that if some voters increase their evaluation of a
particular candidate and raise him or her
in their rankings, the result can be -- paradoxically --
that the candidate actually does worse! For example,
consider an election in which there are four candidates, A,
B, C, D, and 21 electors. Initially, the electors rank the
candidates like this:

7 voters rank: A B C D

6 voters rank: B A C D

5 voters rank: C B A D

3 voters rank: D C B A

In the first round of the tally, the candidate with the
fewest first-place
votes is eliminated, namely D. After D's votes have been
redistributed, the following ranking results:

7 voters rank: A B C

6 voters rank: B A C

5 + 3 = 8 voters rank: C B A

Then B is eliminated, leading to the new ranking:

7 + 6 = 13 voters rank: A C

8 voters rank: C A

Thus A wins the election.

Now suppose that the 3 voters who originally ranked the
candidates D C B A change their mind about A, moving him
from their last place choice to their first place: A D C
B. These voters do not change their evaluation of the other
three candidates, nor do any of the other voters change
their rankings of any of the candidates. But when the votes
are tallied this time, the end result is that B wins. (If
you don't believe this, just work through the tally process
one round at a time. The first round eliminates D, the
second round eliminates C, and the final result is that 10
voters prefer A to B and 11 voters prefer B to A.)

For all the advantages offered by the single transferable
vote system, the fact that a candidate can actually harm
her chances by increasing her voter appeal -- to the point
of losing an election that she would otherwise have won --
leads some mathematicians to conclude that the method
should not be used.

The Borda count has at least two weaknesses. First, it is
easy for blocks of voters to manipulate the outcome. For
example, suppose there are 3 candidates A, B, C and 5
electors, who initially rank the candidates:

3 voters rank: A B C

2 voters rank: B C A

The Borda count for this ranking is as follows:

A: 3x3 + 2x1 = 11

B: 3x2 + 2x3 = 12

C: 3x1 + 2x2 = 7

Thus, B wins. Suppose now that A's supporters realize what
is likely to happen and deliberately change their ranking
from A B C to A C B. The Borda count then changes
to:

A: 11; B: 9; C: 10.

This time, A wins. By putting B lower on their lists, A's
supporters are able to deprive him of the victory he would
otherwise have had.

Of course, almost any method is subject to strategic voting
by a sophisticated electorate, and Borda himself
acknowledged that his system was particularly vulnerable,
commenting: "My scheme is intended only for honest men."
Somewhat more worrying to the student of electoral math is
the fact that the entry of an additional candidate into the
race can dramatically alter the final rankings, even if
that additional candidate has no chance of winning, and
even if none of the voters changes their rankings of the
original candidates. For
example, suppose that there are 3 candidates, A, B, C, in
an election with 7 voters. The voters rank the candidates
as follows:

3 voters rank: C B A

2 voters rank: A C B

2 voters rank: B A C

The Borda count for this ranking is:

A: 13; B: 14; C: 15.

Thus, the candidates final ranking is C B A. Now
candidate
X enters the race, and the voters' ranking becomes:

3 voters rank: C B A X

2 voters rank: A X C B

2 voters rank: B A X C

The new Borda count is:

A: 20; B: 19; C: 18; X: 13.

Thus, the entry of the losing candidate X into the race has
completely reversed the ranking of A, B, and C, giving the
result A B C X.

With even seemingly "sophisticated" vote-tallying methods
having such drawbacks, how are we to decide which is the
best method? Of course, the democratic way to settle the
matter would be to vote on the available systems. But then,
how do
we tally the votes of that election? When it comes
to elections, it seems that even the math used to count the
votes is subject to debate!

But that is no reason not to look for a better way. Many
Americans, and I am one of them, would like to see our
country return to being a world leader in democratic
government, a country that spreads democratic ideals by
example, not by military force. What better place to start
than with the system by which governments are elected?
Let's use our world-leading scientific and technological
know-how to improve the democratic system itself.

The examples I gave above show that there is no easy
solution. But two things are clear. First, as Winston
Churchill said, democracy is a terrible form of government,
but all the rest are much worse. Second, all voting systems
have drawbacks, but plurality voting, our present system,
is the worst, and any of the other systems described
here would surely do a better job of representing the
preferences of the electorate.

For the record, I regularly vote in elections, but I do so
"scientifically," based on information about and past
performance of the candidates; I am not affiliated with any
political party. Part of this column is adapted from my
column of November 2000.

Devlin's Angle is updated at the
beginning
of each month.
Mathematician Keith Devlin (
devlin@csli.stanford.edu) is the
Executive Director of the Center for the
Study of Language and Information at
Stanford University and
The Math Guy on NPR's Weekend Edition.
Devlin's newest book, THE MATH INSTINCT:
Why You're a Mathematical Genius (along with
Lobsters, Birds, Cats, and Dogs), will be
published next spring
by Thunder's Mouth Press.